EE202 Supplementary Materials for Self Study

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EE202 Supplementary Materialsfor Self Study

• Circuit Analysis Using Complex Impedance
• Passive Filters and Frequency Response

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Acknowledgment

• Dr. Furlani and Dr. Liu for lecture slides
• Ms. Colleen Bailey for homework and solution of
  complex impedance
• Textbook Nilsson Riedel, Electric
  Circuits, 8th edition

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Steady-State Circuit Response to Sinusoidal
Excitation - Analysis Using Complex Impedance
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Why Sinusoidal?US Power Grid 60Hz Sinusoidal

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Household Power Line
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Household Circuit Breaker Panel 240V Central
Air 120V Lighting, Plugs, etc.
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Single Frequency Sinusoidal Signal
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Sinusoidal Signal

• Amplitude
• Peak-to-peak
• Root-mean-square
• Frequency
• Angular Frequency
• Period

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Trigonometry Functions
Appendix F
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Other Periodic Waveforms Fundamental and Harmonics
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Resistor Only Circuit

• IV/R, i(t)v(t)/R
• Instantaneous Response

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R-L Circuit
Transient Steady-state
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Phase Shift
Time Delay or Phase Angle ?t / T 2? or
360-degree
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Phasor Complex Number
Z Real(Z)j Imag(Z)
Y Imag(Z)

tan-1(Y/X)
X Real(Z)
Reference
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Complex Number
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Phasor Solution of R-L Circuit
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Observations

• Single Frequency for All Variables
• Phasor Solution of Diff Eq.
• Algebraic equation
• Extremely simple
• Phase
• Delay between variables
• Physical Measurements
• Real part of complex variables
• v RealV i RealI

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Resistor
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Instantaneous Response
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Inductor
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Capacitor
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Impedance in Series
Complex Impedance Resistance, Reactance
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Example
?5000 rad/sec
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Apply ZLj?L, ZC1/j ?C
Zab90j(160-40)90j120sqrt(9021202)expjtan-1(
120/90) 150 ? 53.13 degree I750 ? 30 deg /
150 ? 53.13 deg 5 ? -23.13 deg5exp(-j23.13o)
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Impedance in Parallel
Complex Admittance Conductance, Susceptance
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Example
?200000 rad/sec
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Apply ZLj?L, ZC1/j ?C
Series Use Z Parallel Use Y Y0.2 ? 36.87 deg
Z5 ? -36.87 deg VIZ40 ? -36.87 deg
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Kirchhoffs Laws

• Same
• Current at a Node
• Addition of current vectors (phasors)
• Voltage Around a Loop or Mesh
• Summation of voltage vectors (phasors)

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Delta-T Transformation
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Example
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Delta-T Transformation
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Series, Parallel, Series
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Another Delta-T Transformation
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Thevenin and Norton Transformation
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Thevenin Equivalent Circuit
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Norton Equivalent Circuits
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Voltage divider Vo36.12-j18.84 (V)
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Find VTh
Vx100-I10, VxI(120-j40)-10Vx solve Vx and
I VTH10VxI120784-j288 (V)
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Find ZTh
Calculate Ia Determine Vx Calculate
Ib ZThVT/IT91.2-j38.4 (Ohm)
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Transformer
Time differentiation replaced by ? j?
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AC Sine Wave, Ideal Transformer Voltage and
Current
Power Conserved
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Transformer

• Power Applications
• Convert voltage
• vout(N2/N1) ? vin
• Signal Applications
• Impedance transformation
• Xab(N1/N2)2 ? XL
• Match source impedance with load to maximize
  power delivered to load

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Power Calculations
 
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Frequency Response of Circuits

• Analysis Over a Range of Frequencies
• Amplifier Uniformity
• Filter Characteristics
• Low pass filter
• High pass filter
• Bandpass filter
• Equalizer

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RC Filters
High Pass Low Pass
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Frequency Response

• High Pass
• Low Pass

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Bode Plot

• Log10 (f)
• Compress many orders of magnitude
• Vertical scale
• Linear
• log 10?log10(Vo/Vin)

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Bode Plot

Appendix D, Appendix E
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Summary

• Sinusoidal, Steady-State Analysis
• Complex Impedance
• Zj?L
• Z1/j?C
• All Circuit Analysis Methods Apply
• Analysis Power Systems
• Frequency Response of Circuits

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Homework

• Problem7.pdf
• Solution7.pdf

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• Frequency Response Passive Filters

• 1. Filters
• 2. Low Pass Filter
• High Bass Filter
• Band Pass Filter
• Band Stop Filter
• Series RLC Resonance
• Parallel RLC Resonance

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Tuning a Radio

• Consider tuning in an FM radio station.
• What allows your radio to isolate one station
  from all of the adjacent stations?

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Filters

• A filter is a frequency-selective circuit.
• Filters are designed to pass some frequencies and
  reject others.

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Different Kinds of Filters

• There are four basic kinds of filters
• Low-pass filter - Passes frequencies below a
  critical frequency, called the cutoff frequency,
  and attenuates those above.
• High-pass filter - Passes frequencies above the
  critical frequency but rejects those below.
• Bandpass filter - Passes only frequencies in a
  narrow range between upper and lower cutoff
  frequencies.
• Band-reject filter - Rejects or stops frequencies
  in a narrow range but passes others.

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Active and Passive Filters

• Filter circuits depend on the fact that the
  impedance of capacitors and inductors is a
  function of frequency
• There are numerous ways to construct filters, but
  there are two broad categories of filters
• Passive filters are composed of only passive
  components (resistors, capacitors, inductors) and
  do not provide amplification.
• Active filters typically employ RC networks and
  amplifiers (opamps) with feedback and offer a
  number of advantages.

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Impedance vs. Frequency

• Calculate the impedance of a resistor, a
  capacitor and an inductor at the following
  frequencies.

f 100 Hz 1000 Hz 10,000 Hz
R 100 W 100 W 100 W
ZL j10 W j100 W j1000 W
ZC -j1000 W -j100 W -j10 W
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RC Low-Pass Filter

• A simple low pass filter can be constructed using
  a resistor and capacitor in series.

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Transfer Function H(?)
A Transfer function H(?) is the ratio of the
output to the input
Hv(?) Transfer function for Voltage
Hv(?) describes what the phase shift and
amplitude scaling are.
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Different Kinds of Filters
Ideal frequency response of four types of filters
a) lowpass
b) highpass
d) bandstop
c) bandpass
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Gain

• Any circuit in which the output signal power is
  greater than the input signal
• Power is referred to as an amplifier
• Any circuit in which the output signal power is
  less than the input signal power
• Called an attenuator

• Power gain is ratio of output power to input
  power

Voltage gain is ratio of output voltage to input
voltage
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The Decibel

• Bel is a logarithmic unit that represents a
  tenfold increase or decrease in power

Because the bel is such a large unit, the decibel
(dB) is often used
For voltage
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RC Low-Pass Filter

• For this circuit, we want to compare the output
  (Vo) to the input (Vs)

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RC Low-Pass Filter

• The cutoff frequency is the frequency at which
  the output voltage amplitude is 70.7 of the
  input value (i.e., 3 dB).

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Example

• What is the cutoff frequency for this filter?
• Given

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RL Low-Pass Filter

• A low-pass filter can also be implemented with a
  resistor and inductor.

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RL Low-Pass Filter

• Comparing the output (Vo) to the input (Vs)

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RL Low-Pass Filter

• The cutoff frequency for this circuit design is
  given by

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EXAMPLE RL Low Pass Filter
Design a series RL low-pass filter to filter out
any noise above 10 Hz.
R and L cannot be specified independently to
generate a value for fco 10 Hz or ?co 2?fco.
Therefore, let us choose L100 mH. Then,

• f(Hz) Vs Vo
• 1.0 0.995
• 10 1.0 0.707
• 60 1.0 0.164

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Filters

• Notice the placement of the elements in RC and RL
  low-pass filters.
• What would result if the position of the elements
  were switched in each circuit?

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RC and RL High-Pass Filter Circuits

• Switching elements results in a High-Pass Filter.

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Example

• What resistor value R will produce a cutoff
  frequency of 3.4 kHz with a 0.047 mF capacitor?
  Is this a high-pass or low-pass filter?

This is a High-Pass Filter
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Bandpass Filter
A bandpass filter is designed to pass all
frequencies within a band of frequencies, ?1 lt ?0
lt ?2
B
Bandwidth B of a Filter
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Bandpass Filters
Bandwidth B of a Filter
B
Transfer function
Center frequency
Maximum occurs when
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Example RLC Bandpass Filters
Design a series RLC bandpass filter with cutoff
frequencies f11kHz and f2 10 kHz.
Cutoff frequencies give us two equations but we
have 3 parameters to choose. Thus, we need to
select a value for either R, L, or C and use the
equations to find other values. Here, we choose
C1µF.
f11kHz ? ?1 2?f1 6280 rad/sf2 10 kHz ?
?2 2?f2 62,800 rad/s
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Bandstop Filter
A bandstop filter is designed to stop or
eliminate all frequencies within a band of
frequencies, ?1 lt ?0 lt ?2
Bandwidth B of a Filter
B
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Bandstop Filters
Bandwidth B of a Filter
B
Transfer function
Center frequency
Minimum occurs when
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Formulas for Band Pass and Band Stop Filters
B
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Series Resonance
Resonance is a condition in an RLC circuit in
which the capacitive and inductive reactances are
equal in magnitude, thereby resulting in a purely
resistive impedance.
Input impedance
The series resonant circuit
Resonance occurs when imaginary part is 0
Resonant/center frequency
Resonance occurs when imaginary part is 0
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Resonant/center frequency
Resonance occurs when imaginary part is 0

• At resonance
• The impedance is purely resistive, Z R
• The voltage and the current are in phase, pf1
• The magnitude of transfer function H(w) Z(w) is
  minimum
• The inductor voltage and capacitor voltage can be
  much more than the source voltage

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The current amplitude vs. frequency for the
series resonant circuit
Maximum power
Half of this power is obtained at ?1 and ?2
Half power frequencies
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Quality Factor
The sharpness of the resonance in a resonant
circuit is measured quantitatively by the quality
factor Q
The quality factor of a resonant circuits is the
ratio of its resonant frequency to its bandwidth
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Series Resonance
Relation between Q and bandwidth B
The higher the circuit Q, the smaller the
bandwidth
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Series Resonance
High Q circuit if,
and half power frequency can be approximated as
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Example - Series Resonance

• The problem requires the formula for the
  frequency f.
• Only the inductance and capacitance matter.
• 1/2p (0.25 H 10-7 F)1/2 1 kHz

• Find the resonant frequency in the following
  circuit in Hz.

100 W
250 mH
10 V
0.1 mF
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Series Resonant Circuit
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Parallel Resonance
The parallel-resonant circuit
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Parallel Resonance
Input admittance
Resonance occurs when imaginary part is 0
Resonance occurs when imaginary part is 0
Resonant frequency
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Half power frequency
Bandwidth B
High Q circuit if,
half power frequencies can be approximated as
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Homework Assignment

• Chapter 11 of 8th Edition of TextbookProblems
  31, 36, 41
• Filter Circuit Problems 1, 2

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Filter Analysis
Prob. 1. For the filter circuit below a.
calculate the reactance of the capacitor at 10Hz,
100Hz, 1kHz, 10kHz and 100kHz. b. calculate the
output voltage at each of these frequencies. c.
calculate the cutoff frequency of this
circuit. d. calculate VOUT at the break
frequency. e. plot a graph of output voltage
against frequency on log graph paper.
VIN 10 V?0?
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Solution i) calculate the reactance of the
capacitor at 10Hz, 100Hz, 1kHz, 10kHz and
100kHz. At 10 Hz
At 100 Hz
At 1 kHz
At 10 kHz
At 100 kHz

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ii) calculate the output voltage at each of these
frequencies. At 10 Hz
At 100 Hz
At 1 kHz
At 10 kHz
At 100 kHz
iii) calculate the break frequency of this
circuit.

• calculate VOUT at the break frequency.

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plot a graph of output voltage against frequency
on log graph paper below.
Theoretical Break Frequency.

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Filter Analysis
Prob. 2 Consider the filter circuit below a.
calculate the reactance of the capacitor at 10Hz,
100Hz, 1kHz, 10kHz and 100kHz. b. calculate the
output voltage at each of these frequencies. c.
calculate the cutoff frequency of this
circuit. d. calculate VOUT at the break
frequency. e. plot a graph of output voltage
against frequency on log graph paper.
VIN 10 V?0?
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Solution i) calculate the reactance of the
capacitor at 10Hz, 100Hz, 1kHz, 10kHz and
100kHz. At 10 Hz
At 100 Hz
At 1 kHz
At 10 kHz
At 100 kHz
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ii) calculate the output voltage at each of these
frequencies. At 10 Hz
At 100 Hz
At 1 kHz
At 10 kHz
At 100 kHz
iii) calculate the break frequency of this
circuit.
iv) calculate VOUT at the break frequency.
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iv) plot a graph of output voltage against
frequency on log graph paper below.
Theoretical Break Frequency.
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Example

• What is the cutoff frequency for this filter?
• Given

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Example RL Low Pass Filter
Design a series RL low-pass filter to filter out
any noise above 10 Hz.
R and L cannot be specified independently to
generate a value for wc. Therefore, let us choose
L100 mH. Then,

• F(Hz) V. Vo
• 1.0 0.995
• 10 1.0 0.707
• 60 1.0 0.164

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(1) Identify the following filter circuits as
being low pass, high pass, band pass or band-stop
(4pts).
3) Identify the following filter circuit as being
low pass, high pass, band-pass or band-stop
(2pts).
4) Assume C1µF and the central frequency of this
filter is 2MHz (i.e. 2e6 Hz).

1. Determine the Inductance L (2pts).
2. Determine the bandwidth B if the Q of thecircuit
   is 100 (2pts).
3. Determine the filter angular cutoff frequencies
   ?1 and ?2 (2pts).

Answers (b) and (c) are high-pass (a) and (d)
are low-pass
(2) If the cut-off frequencies for each of the
circuits is 1kHz and the resistance in each
circuit is R1000?, find the values of L or C for
each circuit (8pts).